I have a PC with ASUS motherboard. The Motherboard comes with on board Gigaethernet port Marvell Yukon Gigabit Ethernet 10/100/1000Base-T Adapter, Copper RJ-45.
I've bought a Linksys WRT54GC router to connect a second PC.
The router has 4 10/100 half duplex ports for the LAN side

The Gigaethernet port can be configured as autosensing, 10 half, 10 full, 100 half and 100 full.
I can only make link with 10 full. In autosensing no link can be stablished.
I could found no way to set the speed and duplex in the Linksys router.

Just an idea, but have you tried a different cable? Gigabit Ethernet generally requires Cat6 to work well, otherwise it may throttle down. I haven't got a clue why it wouldn't autosense, though. I'd recommend two things: first, try the same cable in a different PC, and see if that works. If that doesn't work, try a different cable.

1. "The Firmware of the Linksys router is v1.04.6"
I doubt that it would make any difference, but Linksys says that the latest level is 1.05.7, issued April 2007.
If you have trouble getting to that page, start here and click in the Downloads box.
The Marvell site is no help.
A French site has Version 10-series drivers for downloading. But I don't know what a 'Pack version' is.
2. Don is right as to Cat 6 cable for Gigabit speed. However for a short distance, and for 'only' a 100 Megabit speed, Cat 5 should work fine. The cable-swap test was a good idea.
3. Can you change the configuration to try the 'second PC' as the router controller? Or does the other PC have only a wireless adapter? That is, can you try using another computer, one that has a wired adapter?
4. Conversely, can you try a different router on 'this' computer?
5. Presumably you have tried a router hard-reset.
6. If all else fails, is it possible that this happens to be a faulty device? Can you test a replacement?

The second PC has an on board 10/100 FastEthernet port. The second PC has also an ASUS Motherboard.
Again, to establish link I had to hard configure 10 Full in the PC (Control Pannel, System, Hardware, Device Administrator, Network Adapters, Advanced Options, Media Type).
Any way, to link, I had to power Cycle the router.
The router is brand new. I have no second router to test.

Curious. Brand new, but the firmware is something over a year old? If both computers have to use the same reduced setting in order to get communication, then either: a) the router is faulty, or b) both computers (and their OS drivers) are faulty. The easiest thing for you to check is whether the latest firmware helps. (That the device has the latest version of microcode is almost always the first thing that should be attempted. Any request for assistance usually will be predicated upon this condition.) If this doesn't help ---

Presumably there is a warranty on the router. I suggest that you ask your supplier for a replacement. Or that you contact Linksys to see if they have any similar problem reports.

The release notes of the new FW version 1.05.7 for the WRT54GC router only speaks of solving a bug in a particular condition in the wireless service. It seems no help.
Magically, the problem was solved.
In Control Pannel, System, Hardware, Device Administrator, Network adapters, Marvell Yukon ..., Advanced Options, 802.1p Support Port A, it was in Off, I set it in On, Media Type in AutoSense, solved. Why? How knows. Same CAT 5 cable.

The nic has be set as autosense thats the only thing these small home routers understand .There is no way to hardcode a port . If you hardcode a nic you then create a speed/duplex mismatch which will default the router to half duplex and cause errors and slowness.